A SIMPLE LAGRANGIAN MODEL OF MESOSCALE TRANSPORT
OF ADMIXTURES IN THE ATMOSPHERE
GEORGE V.MOSTOVOI, pesquisador-visitante da Universidade de Moscou (RUSSIA).
Fundacao Cearence de Meteorologia e Recursos Hidricos - FUNCEME. Av. Bezerra de Menezes,
1900 - Fortaleza - CE.
ABSTRACT
A mathematical model that allows to estimate instantaneous and time-averaged dispersion of admixture concentration
from a continuous point source in a neutrally stratified atmosphere is described. The transport of the admixture-containing
airborned particles proceeds under the effect of mean and pulsation components of wind velocity. The field of the mean
near-surface wind is reconstructed with a simple hydrostatic atmosphere model. It provides an approximate estimation of the
surface wind vector field from the prescribed large-scale temperature and geopotential fields in the free atmosphere. The
lower limit of a typical horizontal resolution is several kilometers. This diagnostic model is based on primitive
hydrodynamic equations in the sigma-coordinates. Velocity pulsations are described on the basis of a modified Markov
chain which also takes account of spatial correlations of pulsations. Instantaneous patterns of admixture distribution are
presented for the specified surface relief and conditions in the free atmosphere.
1. Introduction
Estimation of air-pollution transport and diffusion is an important task of applied meteorology.
Usually this total effect is called dispersion. A pattern of air pollution associated with a point source
can be updated at the distances of a few up to 30-40 kilometres from a source. Similar horizontal extent
have most plumes generated by large industrial point sources. We shall call it mesoscale air pollution
transport. Sometimes this updating is possible by using more adequate information about mean and
turbulent velocity fields.
Routine large-scale meteorological data from the observational network are commonly available
with a typical horizontal resolution of 100-200 km. Meanwhile a detailed evaluation of a mesoscale airpollution transport requires any interpolation procedure to reconstruct a surface wind velocity field with
a typical horizontal resolution of a few kilometers. The same velocity field can be obtained from a
mesoscale atmosphere model with a proper spatial resolution. In this work the mean near-surface wind
field is reconstructed from a simple mesoscale hydrostatic model 4 . It provides an approximate
estimation of the surface wind vector field from the specified large-scale temperature and geopotential
fields in the free atmosphere.
To describe air-pollution diffusion we use a Lagrangian particle dispersion model. It has definite
advantage over an Eulerian one. Usually the Lagrangian dispersion model is embedded into a
mesoscale one. Some examples of a Lagrangian dispersion model application are documented in 6, 8 .
2. Numerical surface wind field model
To update surface mean wind field over a territory with a typical size of 100x100 km 2 a diagnostic
mesoscale model is used. First it was proposed by C.Mass and D.Dempsey 3 . The model is based on
the primitive equations of motion written in the sigma-coordinates 2 for the surface level   1
 Ln P*
 Z*
u
u
u
u
v
 g
 RT*
 Fx
(1)
t
x
y
x
x
 Ln P*
 Z*
v
v
v
u
v
 g
 RT*
 Fy
(2)
t
x
y
y
y
where u, v, T* , P* are the surface horizontal wind velocity components, the air temperature and the
atmospheric pressure and characterize conditions within a total thickness of the surface air layer.
Z* x , y is the absolute height of the surface, g is the gravitational constant, R is the universal gas
constant, Fx , Fy are the components of a surface friction.




In the right-hand side of (1), (2) terms describing horizontal diffusion are omitted since they are
calculated standardly. The use of the heat conserving equation makes the system (1), (2) closed for the
variables u , v and T* .
One can eliminate the surface atmospheric pressure P* from (1),(2) with the help of the hydrostatic
balance equation. Adopting a linear lapse rate for an air temperature T  T*   Z , it is possible to
obtain the folowing relation from the hydrostatic balance


Z



 

P
g *
dZ
 Ln 
R Z T*   Z
P*

(3)
Where Z and P are the height and the atmospheric pressure for any  -surface lying in the free
atmosphere,     T
z .
The x differencing of (3) gives the following relation for the horizontal pressure gradient
component
T

g
1  Z* 1  Z 1 
(4)

Ln P* 


Ln *
x
R 1   T*  x T  x   x
T
The same expression one can write for the y -component.
Substitution of these relations in the finite-difference analogues of equations (1), (2) provieds total
cancellation of hydrostatically balanced terms for the horizontal pressure gradient components.
Otherwise large computational errors can arise.
To approximate vertical profile of T analytically we use an approach proposed by C.Mass and
D.Dempsey 3 . They introduce the lower atmospheric layer with a constant prescribed thickness and a
horizontally variable  . Above it lies the other layer with an upper boundary Z and a constant value of
 (0.5 C/100m).
The equations (1),(2), the equation for T* and specified relief height Z* x , y were used to estimate
stationary surface wind and temperature fields. A stationary solution of this system is found by
integration, which is proceeded until the normalized difference between the consequent values of both
u and v gets less than 0.01. Boundary conditions, spatial and time finite-differencing are described in
the papers 3, 4 . Model variables are represented on the Arakawa C-grid.
Fig.1 demonstrates an example of the surface wind estimation over the complex terrain with an area
of 60x60 km 2 . It covers partly territory and ocean aquatory surrounding Fortaleza city, State of Ceara.
The land surface topography is shown by isolines. To the SW from the city lies mountain chain with the
peaks up to 900m.
This stationary wind field distribution is found for the following external conditions. Upper top of
the model is equal to 700 mb, the isobaric surface has a constant declination - 0.6 m/2km both along x
and y axes. The conditions are chosen commonly by a user from atmosphere radiosonde data or from
large-scale forecast information. If an additional point with surface wind or air temperature data exists
within the area under consideration, it can be easily assimilated in the model frame.
Proposed model evidently provide an approximate estimation of the surface air flow field in
comparison with more sophisticated nonhydrostatic atmosphere models. At the same time it doesn't
require significant computer resources and has only two prescribed parameters (  and Z in the free
atmosphere over the computation domain). These parameters and the surface relief control air flow
pattern near the surface.















Stationary wind field (fig.1) reproduce particular situation of a relatively weak passat current. It is
often associated with a formation of a mesoscale convective disturbance near the NE litoral zone of
Brazil. A similar meteorological situation was observed 18.06.94.
Fig.1 Stationary surface wind field distribution for the constant NE air flow at 700 mb level.
Isolines show the relief height of the territory, countour interval is 100 m. Outer isoline
is 30 m. Fortaleza town (Ceara State, Brazil) is shown by the dotted area. The square
marks the point source location.
3. Atmosphere dispersion model
For numerical evaluation of atmospheric diffusion pollution we use a Lagrangian particle model.
Instantaneous trajectories coordinates of air polluted particles (volumes) are calculated with a finite
time step t
Xi t  t  Xi t  Ui t  t t
(5)
where Xi is i -coordinate ( i  1, 2 ) of a particle center. Instantaneous wind velocity Ui is decomposed
on time-averaged U i and turbulent Ui' values. The last is calculated from the modified Markov chain,
which is described somewhat in detail in the paper 5 . The calculation requires the estimation of the
 

Lagrangian time scale TLi and the rms of the i velocity turbulent component. Taking into consideration
only neutral stability within the surface atmospheric layer one can use the simple parameterization for
 i 1 : 1  2. 5 U* ,  2  2. 0 U* where U* is the dynamical velocity and calculated from the relation
 / Ln
10
2
2 0.5
U*  0. 4 U1  U 2
Relation (5) reflects a quase-horizontal air polluted particle dispersion along the surface. If we
permit additionally a vertical displacement of particles, then (5) will be associated with horizontal
projection of three-dimensional trajectories. Then a particle concentration field can be interpreted as
vertically-integrated values. In this case a particle volume can be represented only by a horizontal crosssection value ( X1X2 ). Note that if Di is the typical i -dimension of a plume then one can estimate Xi
from any condition, for example like Xi  0.1Di . It garantees adequate spatial resolution of timeaveraged plume concentration structure.
4. Results
The model run example has been done with the determined parameters:  X1   X2  25m, t = 6s,
1
TL  TL2  60s . Fig.2 shows the instantaneous Lagrangian particle distribution generated by a continous
point source. The particles centers are marked by dots. The distribution pattern corresponds to the 30
min emission time. Dashed lines represent the regular grid in the points of which the mesoscale wind
field is evaluated. The point source location is marked by the square on Fig.1. The specified point
source position corresponds to the polluted particles dispersion in a weak and nondivergent air flow
(average wind velocity is slightly below 3 m/s). Fig.3 shows Lagrangian particles distribution produced
by the instantaneous point source which position is marked by the circle.
The information about current particle diffusion parameters is outputed each time step on the
computer screen. The parameters are: the number of time steps passed from switch on of a source
(time), the time step (dt, s), the rms turbulent velocity components (sigma(v), m/s), the Lagrange time
Fig. 2 Instantaneous distribution of the Lagrangian polluted particles (dots) emitted by the
continuous point source. Emission time is 30 min. Dashed lines shows the regular grid in
the points of which the surface wind field is estimated.
scale (TL, s) and the mean velocity components (ux, uy, m/s).
Note that purely random component of the particle velocity is generated with a special procedure
7 to have the certain correlation interval proportional to U i TLi . Thus the dispersion particle model
takes account of the atmospheric boundary layer eddy structure. This feature of the atmospheric
diffusion provides well-noticed alternation of non- and dense parts of particle plume. The similar
intermittency of regions filled with smoke particles and clean ambient areas is an important visual
pecularity of natural smoke plumes.
5. References
1. Hall C.D. The simulation of particle motion in the atmosphere by a numerical random-walk model. Q.J.Roy.Met. Soc., 1975, vol. 101, N428.
2. Holton J.R. An Introduction to Dynamic Meteorology. 1979, New York, AP.
3. Mass C.F., Dempsey D.P. A one-level mesoscale model suitable for nowcasting and short-term forecasting in
regions of complex terrain. - Nowcasting II - Proceedings of the Second International Symposium of
Nowcasting, Norrkoping, Sweden 3-7 Sept., 1984, ESA.
4. Mostovoi G.V. A simple hydrodynamic model for surface wind field diagnosis. - Met. & Hydrol., 1993, N3,
p. 14-21 (in Russian).
5. Mostovoi G.V. A simple Lagrangian model of mesoscale transport of admixtures in the atmosphere. - Met. &
Hydrol. 1993, N5, p.29-35 (in Russian).
6. Pielke R. et al. The use of a mesoscale numerical model for evaluations of pollutant transport and diffusion in
coastal regions and over irregular terrain. - Bull. Amer. Met. Soc., 1983, vol. 64, N3.
7. Shimanuki A., Nomura Y. Numerical simulation of instantaneous images of the smoke released from a
chimney. - J.Met.Soc. of Japan, 1991, vol.69, N2.
Fig. 3 Lagrangian particles distribution (dots) produced by the instantaneous point source. Diffusion time
is 30 min, dashed lines show the regular grid in the points of which the surface wind field is
estimated.